Finite amplitude convection between stress-free boundaries; Ginzburg–Landau equations and modulation theory

The stability theory for rolls in stress-free convection at finite Prandtl number is affected by coupling with low wavenumber two-dimensional mean-flow modes. In this work, a set of modified Ginzburg–Landau equations describing the onset of convection is derived which accounts for these additional modes. These equations can be used to extend the modulation equations of Zippelius & Siggia describing the breakup of rolls, bringing their stability theory into agreement with the results of Busse & Bolton.